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Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to: three points (denoted PPP, generally 1 solution)
The line segments AB and CD are perpendicular to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Although many authors use the two terms perpendicular and orthogonal interchangeably, the term perpendicular is more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations ...
To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the ...
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (radians), or one of the vectors is zero. [4] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.
Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle). [9] Two angles are called complementary if their sum is a right angle. [10] Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a ...
Then the vertex of this angle is X and traces out the pedal curve. As the angle moves, its direction of motion at P is parallel to PX and its direction of motion at R is parallel to the tangent T = RX. Therefore, the instant center of rotation is the intersection of the line perpendicular to PX at P and perpendicular to RX at R, and this point ...
Carnot's theorem: if three perpendiculars on triangle sides intersect in a common point F, then blue area = red area. Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection.
Any two pairs of angles are congruent, [4] which in Euclidean geometry implies that all three angles are congruent: [a] If ∠ BAC is equal in measure to ∠ B'A'C', and ∠ ABC is equal in measure to ∠ A'B'C', then this implies that ∠ ACB is equal in measure to ∠ A'C'B' and the triangles are similar.